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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2502.14723 |
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| _version_ | 1866910837345615872 |
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| author | Christov, Ognyan Hakkaev, Sevdzhan Oh, Seungly Stefanov, Atanas G. |
| author_facet | Christov, Ognyan Hakkaev, Sevdzhan Oh, Seungly Stefanov, Atanas G. |
| contents | We analyze the Drinfeld-Sokolob-Wilson system, which features a dispersive, KdV type evolution with a dispersionless conservation law. We establish well-posedness with low regularity initial data $L^2({\mathbb T})\times L^2({\mathbb T})$ for the Cauchy problem on periodic background, which is then extrapolated to global solutions, due to $L^2$ conservation law. We also establish a dynamically more relevant result, namely a global persistence of solutions with (large) initial data in $H^1({\mathbb T})\times L^2({\mathbb T})$. This is obtained by following a more sophisticated approach, specifically the method of normal forms.
Finally, for a fixed period $L$, we construct an explicit one parameter family of periodic waves, see \eqref{2.16} below. We show that they are all spectrally unstable with respect to co-periodic perturbations. Specifically, we show that the Hamiltonian instability index is equal to one, which identifies the instability as a single positive growing mode. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_14723 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Dynamics of the Drinfeld-Sokolov-Wilson system: well-posedness and (in)stability of the traveling waves Christov, Ognyan Hakkaev, Sevdzhan Oh, Seungly Stefanov, Atanas G. Analysis of PDEs 35Q53, 76B25 We analyze the Drinfeld-Sokolob-Wilson system, which features a dispersive, KdV type evolution with a dispersionless conservation law. We establish well-posedness with low regularity initial data $L^2({\mathbb T})\times L^2({\mathbb T})$ for the Cauchy problem on periodic background, which is then extrapolated to global solutions, due to $L^2$ conservation law. We also establish a dynamically more relevant result, namely a global persistence of solutions with (large) initial data in $H^1({\mathbb T})\times L^2({\mathbb T})$. This is obtained by following a more sophisticated approach, specifically the method of normal forms. Finally, for a fixed period $L$, we construct an explicit one parameter family of periodic waves, see \eqref{2.16} below. We show that they are all spectrally unstable with respect to co-periodic perturbations. Specifically, we show that the Hamiltonian instability index is equal to one, which identifies the instability as a single positive growing mode. |
| title | Dynamics of the Drinfeld-Sokolov-Wilson system: well-posedness and (in)stability of the traveling waves |
| topic | Analysis of PDEs 35Q53, 76B25 |
| url | https://arxiv.org/abs/2502.14723 |